Question

The distance between two locations, A and B, is calculated using a third location C at a distance of 15 miles from location B. If ∠B = 105° and ∠C = 20°, what is the distance, to the nearest tenth of a mile, between locations A and B?

Answer 1

Distance between A and B is 5.7 miles

Explanation:

In the given triangle ABC

∠B = 105° and ∠C = 20°

and distance between locations B and C = 15 miles.

We have to determine the distance between locations A and B.

In triangle ABC

∠A = 180 - (∠B + ∠C) = 180 - (105 + 20) = 180 - 125 = 55°

`(sinA)/(a)=(sinB)/(b)=(sinC)/(c)`

Now we place the values of angles and distance BC in the formula

`(sinA)/(a)=(sinC)/(c)`

`(sin55)/(15)=(sin20)/(c)`

`(0.8192)/(15)=(0.342)/(c)`

`(0.342)/(c)=.0546`

`c=(0.342)/(.0546)=6.26`

Therefore distance between locations A and B is 6.3 miles

Answer 2

We will first calculate ∠A:
∠∠A = 180° - ( 105° + 20 ° ) = 180° - 125° = 55°
Then we will use the Sine Law:
15 / sin 55° = AB / sin 20°
15 / 0.81915 = AB / 0.342
AB · 0.81915 = 15 · 0.342
AB · 0.81915 = 5.13
AB = 5.13 : 0.81915
AB = 6.26 ≈ 6.3 miles
Answer: The distance between the locations A and B is 6.3 miles.